Matrix mechanics

Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. It was the first conceptually autonomous and logically consistent formulation of quantum mechanics. Its account of quantum jumps supplanted the Bohr model's electron orbits. It did so by interpreting the physical properties of particles as matrices that evolve in time. It is equivalent to the Schrödinger wave formulation of quantum mechanics, as manifest in Dirac's bra–ket notation.
In some contrast to the wave formulation, it produces spectra of operators by purely algebraic, ladder operator methods. Relying on these methods, Wolfgang Pauli derived the hydrogen atom spectrum in 1926, before the development of wave mechanics.

Development of matrix mechanics

In 1925, Werner Heisenberg, Max Born, and Pascual Jordan formulated the matrix mechanics representation of quantum mechanics.

Epiphany at Heligoland

In 1925 Werner Heisenberg was working in Göttingen on the problem of calculating the spectral lines of hydrogen. By May 1925 he began trying to describe atomic systems by observables only. On June 7, after weeks of failing to alleviate his hay fever with aspirin and cocaine, Heisenberg left for the pollen-free North Sea island of Heligoland. While there, in between climbing and memorizing poems from Goethe's West-östlicher Diwan, he continued to ponder the spectral issue and eventually realised that adopting non-commuting observables might solve the problem. He later wrote:

It was about three o' clock at night when the final result of the calculation lay before me. At first I was deeply shaken. I was so excited that I could not think of sleep. So I left the house and awaited the sunrise on the top of a rock.

The three fundamental papers

After Heisenberg returned to Göttingen, he showed Wolfgang Pauli his calculations, commenting at one point:

Everything is still vague and unclear to me, but it seems as if the electrons will no more move on orbits.

On July 9 Heisenberg gave the same paper of his calculations to Max Born, saying that "he had written a crazy paper and did not dare to send it in for publication, and that Born should read it and advise him" prior to publication. Heisenberg then departed for a while, leaving Born to analyse the paper.
In the paper, Heisenberg formulated quantum theory without sharply-defined electron orbits, directly advocating for a re-interpretation of quantum theory that only focused on experimental observables like frequencies and transition probabilities.
Before Heisenberg's paper, Hendrik Kramers had calculated the relative intensities of spectral lines in the Sommerfeld model by interpreting the Fourier coefficients of the orbits as intensities. But his answer, like all other calculations in the old quantum theory, was only correct for large orbits.
Heisenberg, after a collaboration with Kramers, began to believe that the transition probabilities describing quantum transitions would need a new interpretation different from classical mechanics because Heisenberg believed that the frequencies that should appear in a series describing the position of the electron should only be the ones that are experimentally observed in quantum transitions, not the complete set of spatial frequencies that come from making a traditional Fourier series of classical orbits.
The quantities in Heisenberg's original formulation involved a series that described position as a series of "virtual oscillators" with two indices, with the two indices representing the initial and final states of a quantum transition. Rather than following the multiplication rule as expected from multiplying Fourier series, Heisenberg formed a non-commutative multiplication rule to ensure that multiplying position states would preserve the frequencies that are only found in the quantum transitions.
When Born read the paper, he recognized the formulation, particularly the non-commutative multiplication rule, as one which could be transcribed and extended to the systematic language of matrices, which he had learned from his study under Jakob Rosanes at Breslau University. Born, with the help of his assistant and former student Pascual Jordan, began immediately to make the transcription and extension, and they submitted their results for publication; the paper was received for publication just 60 days after Heisenberg's paper.
A follow-on paper was submitted for publication before the end of the year by all three authors.
Up until this time, matrices were seldom used by physicists; they were considered to belong to the realm of pure mathematics, thus requiring Born and Jordan's paper to introduce matrix algebra to physicists unaware of their use. Gustav Mie had used them in a paper on electrodynamics in 1912 and Born had used them in his work on the lattices theory of crystals in 1921. While matrices were used in these cases, the algebra of matrices with their multiplication did not enter the picture as they did in the matrix formulation of quantum mechanics.
Born, however, had learned matrix algebra from Rosanes, as already noted, but Born had also learned Hilbert's theory of integral equations and quadratic forms for an infinite number of variables as was apparent from a citation by Born of Hilbert's work Grundzüge einer allgemeinen Theorie der Linearen Integralgleichungen published in 1912.
Jordan, too, was well equipped for the task. For a number of years, he had been an assistant to Richard Courant at Göttingen in the preparation of Courant and David Hilbert's book Methoden der mathematischen Physik I, which was published in 1924. This book, fortuitously, contained a great many of the mathematical tools necessary for the continued development of quantum mechanics.
In 1926, John von Neumann became assistant to David Hilbert, and he would coin the term Hilbert space to describe the algebra and analysis which were used in the development of quantum mechanics.
A linchpin contribution to this formulation was achieved in Dirac's reinterpretation/synthesis paper of 1925, which invented the language and framework usually employed today, in full display of the noncommutative structure of the entire construction.

Heisenberg's reasoning

Before matrix mechanics, the old quantum theory described the motion of a particle by a classical orbit, with well defined position and momentum with the restriction that the time integral over one period of the momentum times the velocity must be a positive integer multiple of the Planck constant as described by the Sommerfeld-Wilson quantization condition
While this restriction correctly selects orbits with the
right energy values the old quantum formalism did not describe time dependent processes, such as the emission or absorption of radiation.
When a classical particle is weakly coupled to a radiation field, so that the radiative damping can be neglected, it will emit radiation in a pattern that repeats itself every orbital period. The frequencies that make up the outgoing wave are then integer multiples of the orbital frequency, and this is a reflection of the fact that is periodic, so that its Fourier representation has frequencies only.
The coefficients are complex numbers. The ones with negative frequencies must be the complex conjugates of the ones with positive frequencies, so that will always be real,
A quantum mechanical particle, on the other hand, cannot emit radiation continuously; it can only emit photons. Under the Bohr model, for a quantum particle starting in quantum number that then emits a photon by transitioning to orbit number the energy of the photon is that gives a photon of frequency is
For large and but with relatively small, Bohr's correspondence principle expects the same classical frequencies
In the formula above, is the classical period of either orbit or orbit since the difference between them is higher order in But for small and or if is large, the frequencies are not integer multiples of any single frequency.
Since in classical mechanics, the frequencies that the particle emits are the same as the frequencies in the Fourier description of its motion, Heisenberg inferred that in the time-dependent description of the particle, there should be something oscillating with frequency Heisenberg called this quantity and demanded that it should reduce to the classical Fourier coefficients in the classical limit. For large values of and but with relatively small,
is the Fourier coefficient of the classical motion at orbit Since has opposite frequency to the condition that is real becomes
By definition, only has the frequency so its time evolution is may be described as:
This is the original form of Heisenberg's equation of motion.
Given two arrays and describing two physical quantities, when modeling each as classical Fourier series, it is expected that their multiplication should also result in a new frequency as part of a new Fourier series. Whilst the Fourier coefficients of the product of two quantities is the convolution of the Fourier coefficients of each one separately, Heisenberg changed the multiplication rule to ensure that when multiplying each component, the new frequencies would only correspond to frequencies that already existed in the quantum orbit:
Born noticed that this is the law of matrix multiplication, so that the position, the momentum, the energy, all the observable quantities in the theory, are interpreted as matrices. Under this multiplication rule, the product depends on the order: is different from
The matrix is a complete description of the motion of a quantum mechanical particle. Because the frequencies in the quantum motion are not multiples of a common frequency, the matrix elements cannot be interpreted as the Fourier coefficients of a sharp classical trajectory. Nevertheless, as matrices, and satisfy the classical equations of motion; also see Ehrenfest's theorem, below.